Invariants
Level: | $60$ | $\SL_2$-level: | $30$ | Newform level: | $80$ | ||
Index: | $72$ | $\PSL_2$-index: | $72$ | ||||
Genus: | $1 = 1 + \frac{ 72 }{12} - \frac{ 16 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
Cusps: | $4$ (none of which are rational) | Cusp widths | $6^{2}\cdot30^{2}$ | Cusp orbits | $2^{2}$ | ||
Elliptic points: | $16$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | $0$ | ||||||
$\Q$-gonality: | $2$ | ||||||
$\overline{\Q}$-gonality: | $2$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 30D1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.72.1.397 |
Level structure
$\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}19&40\\34&53\end{bmatrix}$, $\begin{bmatrix}33&35\\46&21\end{bmatrix}$, $\begin{bmatrix}37&15\\0&11\end{bmatrix}$, $\begin{bmatrix}42&55\\55&42\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | none in database |
Cyclic 60-isogeny field degree: | $24$ |
Cyclic 60-torsion field degree: | $384$ |
Full 60-torsion field degree: | $30720$ |
Jacobian
Conductor: | $2^{4}\cdot5$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 80.2.a.b |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 5 x^{2} + z w $ |
$=$ | $5 y^{2} + 5 z^{2} + 2 z w + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} - 2 x^{2} z^{2} + y^{2} z^{2} + 5 z^{4} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
$\displaystyle X$ | $=$ | $\displaystyle x$ |
$\displaystyle Y$ | $=$ | $\displaystyle y$ |
$\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{5}w$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle -\frac{(5z^{6}-10z^{3}w^{3}+w^{6})^{3}}{w^{3}z^{15}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
30.36.0.f.1 | $30$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
60.36.0.i.2 | $60$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
60.36.1.ga.1 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
60.144.9.u.1 | $60$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
60.144.9.z.1 | $60$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
60.144.9.cw.1 | $60$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
60.144.9.cz.1 | $60$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
60.144.9.io.2 | $60$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
60.144.9.it.1 | $60$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
60.144.9.jk.1 | $60$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2^{2}$ |
60.144.9.jo.1 | $60$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
60.216.9.bq.1 | $60$ | $3$ | $3$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
60.288.13.sq.1 | $60$ | $4$ | $4$ | $13$ | $1$ | $1^{6}\cdot2^{3}$ |
60.360.21.df.1 | $60$ | $5$ | $5$ | $21$ | $4$ | $1^{8}\cdot2^{6}$ |
120.144.9.iuv.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.iwe.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.kkt.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.klo.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.szs.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.tbi.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.tgl.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.thu.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
180.216.13.hs.2 | $180$ | $3$ | $3$ | $13$ | $?$ | not computed |
300.360.21.x.1 | $300$ | $5$ | $5$ | $21$ | $?$ | not computed |